Dynamical System

A dynamical system is a system in which a function describes how a point moves through an ambient space over time, as in a parametric curve; examples include mathematical models for the swing of a clock pendulum, the flow of water in a pipe, the random motion of particles in the air, and the number of fish in a lake each spring. The most general definition unifies concepts such as ordinary differential equations and ergodic theory by allowing different choices of space and of how time is measured, with time represented by integers, real or complex numbers, or a more general algebraic object. At any given time a dynamical system has a state representing a point in its state space, and an evolution rule, a function describing what future states follow from the current one, which may be deterministic or, in stochastic systems, influenced by random events. The study of dynamical systems, called dynamical systems theory, applies to fields including physics, biology, chemistry, engineering, economics and medicine, and underlies chaos theory, logistic map dynamics and bifurcation theory.